SRC/sstebz.f(3)

NAME
SRC/sstebz.f

SYNOPSIS

  Functions/Subroutines
subroutine sstebz (range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
SSTEBZ

Function/Subroutine Documentation

  subroutine sstebz (character range, character order, integer n, real vl, real vu, integer il, integer iu, real abstol, real, dimension( * ) d, real, dimension( * ) e, integer m, integer nsplit, real, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSTEBZ

Purpose:

    !>
    !> SSTEBZ computes the eigenvalues of a symmetric tridiagonal
    !> matrix T.  The user may ask for all eigenvalues, all eigenvalues
    !> in the half-open interval (VL, VU], or the IL-th through IU-th
    !> eigenvalues.
    !>
    !> To avoid overflow, the matrix must be scaled so that its
    !> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
    !> accuracy, it should not be much smaller than that.
    !>
    !> See W. Kahan , Report CS41, Computer Science Dept., Stanford
    !> University, July 21, 1966.
    !>

Parameters
    RANGE

    !>          RANGE is CHARACTER*1
    !>          = 'A': ()   all eigenvalues will be found.
    !>          = 'V': () all eigenvalues in the half-open interval
    !>                           (VL, VU] will be found.
    !>          = 'I': () the IL-th through IU-th eigenvalues (of the
    !>                           entire matrix) will be found.
    !>

    ORDER

    !>          ORDER is CHARACTER*1
    !>          = 'B': () the eigenvalues will be grouped by
    !>                              split-off block (see IBLOCK, ISPLIT) and
    !>                              ordered from smallest to largest within
    !>                              the block.
    !>          = 'E': ()
    !>                              the eigenvalues for the entire matrix
    !>                              will be ordered from smallest to
    !>                              largest.
    !>

    N

    !>          N is INTEGER
    !>          The order of the tridiagonal matrix T.  N >= 0.
    !>

    VL

    !>          VL is REAL
    !>
    !>          If RANGE='V', the lower bound of the interval to
    !>          be searched for eigenvalues.  Eigenvalues less than or equal
    !>          to VL, or greater than VU, will not be returned.  VL < VU.
    !>          Not referenced if RANGE = 'A' or 'I'.
    !>

    VU

    !>          VU is REAL
    !>
    !>          If RANGE='V', the upper bound of the interval to
    !>          be searched for eigenvalues.  Eigenvalues less than or equal
    !>          to VL, or greater than VU, will not be returned.  VL < VU.
    !>          Not referenced if RANGE = 'A' or 'I'.
    !>

    IL

    !>          IL is INTEGER
    !>
    !>          If RANGE='I', the index of the
    !>          smallest eigenvalue to be returned.
    !>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
    !>          Not referenced if RANGE = 'A' or 'V'.
    !>

    IU

    !>          IU is INTEGER
    !>
    !>          If RANGE='I', the index of the
    !>          largest eigenvalue to be returned.
    !>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
    !>          Not referenced if RANGE = 'A' or 'V'.
    !>

    ABSTOL

    !>          ABSTOL is REAL
    !>          The absolute tolerance for the eigenvalues.  An eigenvalue
    !>          (or cluster) is considered to be located if it has been
    !>          determined to lie in an interval whose width is ABSTOL or
    !>          less.  If ABSTOL is less than or equal to zero, then ULP*|T|
    !>          will be used, where |T| means the 1-norm of T.
    !>
    !>          Eigenvalues will be computed most accurately when ABSTOL is
    !>          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          The n diagonal elements of the tridiagonal matrix T.
    !>

    E

    !>          E is REAL array, dimension (N-1)
    !>          The (n-1) off-diagonal elements of the tridiagonal matrix T.
    !>

    M

    !>          M is INTEGER
    !>          The actual number of eigenvalues found. 0 <= M <= N.
    !>          (See also the description of INFO=2,3.)
    !>

    NSPLIT

    !>          NSPLIT is INTEGER
    !>          The number of diagonal blocks in the matrix T.
    !>          1 <= NSPLIT <= N.
    !>

    W

    !>          W is REAL array, dimension (N)
    !>          On exit, the first M elements of W will contain the
    !>          eigenvalues.  (SSTEBZ may use the remaining N-M elements as
    !>          workspace.)
    !>

    IBLOCK

    !>          IBLOCK is INTEGER array, dimension (N)
    !>          At each row/column j where E(j) is zero or small, the
    !>          matrix T is considered to split into a block diagonal
    !>          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
    !>          block (from 1 to the number of blocks) the eigenvalue W(i)
    !>          belongs.  (SSTEBZ may use the remaining N-M elements as
    !>          workspace.)
    !>

    ISPLIT

    !>          ISPLIT is INTEGER array, dimension (N)
    !>          The splitting points, at which T breaks up into submatrices.
    !>          The first submatrix consists of rows/columns 1 to ISPLIT(1),
    !>          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
    !>          etc., and the NSPLIT-th consists of rows/columns
    !>          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
    !>          (Only the first NSPLIT elements will actually be used, but
    !>          since the user cannot know a priori what value NSPLIT will
    !>          have, N words must be reserved for ISPLIT.)
    !>

    WORK

    !>          WORK is REAL array, dimension (4*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (3*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  some or all of the eigenvalues failed to converge or
    !>                were not computed:
    !>                =1 or 3: Bisection failed to converge for some
    !>                        eigenvalues; these eigenvalues are flagged by a
    !>                        negative block number.  The effect is that the
    !>                        eigenvalues may not be as accurate as the
    !>                        absolute and relative tolerances.  This is
    !>                        generally caused by unexpectedly inaccurate
    !>                        arithmetic.
    !>                =2 or 3: RANGE='I' only: Not all of the eigenvalues
    !>                        IL:IU were found.
    !>                        Effect: M < IU+1-IL
    !>                        Cause:  non-monotonic arithmetic, causing the
    !>                                Sturm sequence to be non-monotonic.
    !>                        Cure:   recalculate, using RANGE='A', and pick
    !>                                out eigenvalues IL:IU.  In some cases,
    !>                                increasing the PARAMETER  may
    !>                                make things work.
    !>                = 4:    RANGE='I', and the Gershgorin interval
    !>                        initially used was too small.  No eigenvalues
    !>                        were computed.
    !>                        Probable cause: your machine has sloppy
    !>                                        floating-point arithmetic.
    !>                        Cure: Increase the PARAMETER ,
    !>                              recompile, and try again.
    !>

Internal Parameters:

    !>  RELFAC  REAL, default = 2.0e0
    !>          The relative tolerance.  An interval (a,b] lies within
    !>           if  b-a < RELFAC*ulp*max(|a|,|b|),
    !>          where  is the machine precision (distance from 1 to
    !>          the next larger floating point number.)
    !>
    !>  FUDGE   REAL, default = 2
    !>          A  to widen the Gershgorin intervals.  Ideally,
    !>          a value of 1 should work, but on machines with sloppy
    !>          arithmetic, this needs to be larger.  The default for
    !>          publicly released versions should be large enough to handle
    !>          the worst machine around.  Note that this has no effect
    !>          on accuracy of the solution.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 270 of file sstebz.f.

Author

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